Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups Andrew F. Misseldine Brigham Young University - Provo Follow this and additional works at: Part of the Mathematics Commons BYU ScholarsArchive Citation Misseldine, Andrew F., "Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups" (2014). All Theses and Dissertations This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu.

2 Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups Andrew Frank Misseldine A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Stephen Humphries, Chair Darrin Doud Tyler Jarvis William Lang Pace Nielsen Department of Mathematics Brigham Young University May 2014 Copyright c 2014 Andrew Frank Misseldine All Rights Reserved

3 ABSTRACT Algebraic and Combinatorial Properties of Schur Rings over Cyclic Groups Andrew Frank Misseldine Department of Mathematics, BYU Doctor of Philosophy In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of Schur rings over cyclic p-groups. Keywords: Schur ring, cyclic group, group ring, primitive idempotent, cyclotomic field, Wedderburn decomposition, representation theory, Galois theory, combinatorics

4 ACKNOWLEDGMENTS I first thank my advisor, Dr. Stephen P. Humphries, for the large amount of time, support, wisdom, and patience he gave to me throughout the development of this dissertation. This dissertation would not be what it is without him. I also thank the other members of the committee for their assistance. I thank Brent Kerby, a former student of my advisor, whose MAGMA code, available in [11], was an invaluable tool in the preparation of this dissertation. I thank Dr. Michael Barrus for his suggestions and instructions to me on many topics of combinatorics. I could not have finished Chapter 5 without his guidance. Finally, I also thank my dear wife for all the love and support she has given me and allowing me to go to school for -many years.

5 Contents List of Tables List of Figures vi vii 1 Introduction 1 2 Schur Rings Group Rings Schur Rings Cayley Maps Pre-Schur Rings Primitive Idempotents of Schur Rings over Cyclic Groups Primitive Idempotents of Semilattice Algebras Primitive Idempotents of Group Algebras Primitive Idempotents of Schur Rings Classification of Schur Rings over Cyclic Groups A Correspondence Between Schur Rings and Cyclotomic Fields Wedderburn Decompositions of Schur Rings over Cyclic Groups Schur Rings over Cyclic Groups of Prime Power Order Counting Schur Rings over Cyclic Groups Counting Schur Rings Over Cyclic p-groups, p odd Counting Schur Rings Over Cyclic p-groups, p even A Semisimple Algebras 102 B Orbit Algebras and Cyclotomic Fields 108 C Lattices of Cyclotomic Fields 112 iv

6 D Magma Code 122 Bibliography 148 Index 151 v

7 List of Tables 2.1 Multiplication Table for F [G] Multiplication Table for the Schur Ring in Example Multiplication Table for the Schur Ring in Example Multiplication Table of Z(Q[Q 8 ]) Multiplication Table for the Schur Ring in Example The first several values of Ω(n, k) The Triangular Array of c jk Coefficients Catalan s Triangle The first several Ω-polynomials Number of Schur Rings over Z p k The Triangular Array of s jk Coefficients Super-Catalan s Triangle Number of Schur Rings over Z 2 n vi

8 List of Figures C.1 The Lattice of Subfields of Q(ζ 81 ) C.2 The Lattice of Subfields of Q(ζ 5 4) C.3 The Lattice of Subfields of Q(ζ 7 4) C.4 The Lattice of Subfields of Q(ζ 8 ) C.5 The Lattice of Subfields of Q(ζ 16 ) C.6 The Lattice of Subfields of Q(ζ 64 ) vii

9 Chapter 1. Introduction In Finite Group Representation Theory, the group algebra provides a valuable tool, as well as many of its subalgebras. The group algebra is a special example of a class of algebras called Schur rings, as well as many other subalgebras such as the center of the group algebra or double coset subalgebras. In many ways, Schur rings generalize the idea of group algebras and capture many of the critical subalgebras. Loosely speaking, a Schur ring is a subalgebra of the group algebra which is spanned by a partition of the finite group and satisfies other properties (see Definition 2.10). Schur rings were originally developed by Schur and Wielandt in the first half of the 20th century and were used to study permutation groups. In particular, certain properties of a Schur ring can determine properties of a related permutation group, such as 2-transitivity or primitivity. In later decades applications of Schur rings have emerged in combinatorics, graph theory, and design theory [12, 19], such as the study of association schemes. Both Wieldant s and Scott s monographs [35, Chapter 4 ], [29, Chapter 13] provide an introduction to the subject of Schur rings. Muzychuk and Ponomarenko also offer a recent survey of Schur rings in [22]. Schur rings over cyclic groups have been extremely useful in the study of circulant graphs. For this reason, Schur rings over cyclic groups have been well studied and a surge of papers emerged in the 1980 s and 1990 s, many of which are included in the bibliography, seeking a complete structure theorem of Schur rings over cyclic groups. This was eventually obtained by Leung and Man around the mid-1990 s (Theorem 2.66). The purpose of this dissertation is to provide even more understanding about Schur rings over cyclic groups. When possible, we will try to make the arguments general, but ultimately the focus will be on Schur rings over cyclic groups. There are two main questions about these Schur rings which this dissertation will answer, one algebraic and one combinatorial. First, what are the primitive idempotents of Schur rings over cyclic groups? Second, how many Schur rings over cyclic groups are there? In Chapter 2, we begin with the basics of Schur rings and their generalizations. This chapter surveys many of the elementary properties of Schur rings with proofs and references 1

10 to the original papers in the literature. Here many detailed constructions of Schur rings with examples are included, including new constructions introduced by the author. Chapter 2 provides the fundamental prerequisites for the rest of the dissertation. In Chapter 3, we introduce a method to construct a complete system of orthogonal central idempotents in Schur rings. This method generalizes methods used by others to build primitive central idempotents in rational group algebras which avoids the use of characters. When the group is cyclic, we will prove that these central idempotents are necessarily primitive (Theorem 3.32). This provides an answer to the algebraic question. Prior to the completion of this dissertation, the contents of this chapter were published in [20]. In Chapter 4, we construct a representation of Schur rings over cyclic groups inside a cyclotomic field. This allows us to use Galois theory to study these Schur rings. One consequence of this work is that we have provided another (simpler) proof of the Leung- Man Classification Theorem of Schur rings over cyclic groups, at least when the order of the group is a power of a prime (Theorem 4.36). A second consequence of this representation is that we provide a Wedderburn decomposition for Schur rings over cyclic groups with rational coefficients (Theorem 4.17). A final consequence is given in Chapter 5, where we give formulas to count the number of Schur rings over specific cyclic groups (Theorem 5.11 and Theorem 5.19). This provides an answer to the combinatorial question. A few appendices are included for the convenience of the reader. Appendix A provides a basic introduction, including proofs, to semisimple rings and their idempotents. Appendix B includes a quick treatment of subalgebras fixed under groups of automorphisms. These type of subalgebras arise often in Galois theory and with Schur rings, so we have included a few general results. Appendix C contains a description of the lattice of subfields of cyclotomic fields. The shape of these lattices will be useful in Chapter 4 and especially in Chapter 5. Finally, Appendix D includes the author s MAGMA code used for the calculations made in Chapter 5. All computations made in preparation of this dissertation were accomplished using the computer softwares Maple and Magma [1]. Before closing this introduction, we will declare some common notation used throughout the paper. Unless otherwise specified, G will denote a finite group and F a field with 2

11 characteristic zero. Let Z n = z n denote the cyclic group of order n. Since each subgroup of Z n is necessarily cyclic and is uniquely determined by its order, for each d n, we will denote the unique subgroup of Z n of order d as Z d. Throughout, let ζ n = e 2πi/n C and let K n = Q(ζ n ). Let L n denote the lattice of subfields of K n. Let G n denote the Galois group G(K n /Q). When the context is clear, subscripts may be omitted. All algebras are associative with unity. Subalgebras will have the same unity as the over algebra. If A is an F -algebra, let Z(A) denote the center of A. Other commonly used notation and vocabulary will be introduced with boldface font. A list of notation can be found in the index. 3

12 Chapter 2. Schur Rings In this chapter we begin our study of Schur rings, which are subalgebras of a group algebra afforded by certain partitions of the group. The fact that they are subalgebras of group algebras implies that Schur rings inherit many properties from the group algebra and in many ways behave like group algebras. In fact, every group algebra has a Schur ring structure, and hence the theory of Schur rings may be seen as a generalization of the theory of group algebras or of groups themselves. The purpose of this chapter will be to introduce and familiarize the reader with the fundamental definitions, properties, and examples of Schur rings and to prepare the reader for the more difficult theory which fills the remainder of this dissertation. Section 2.1 begins with group rings themselves and provides definitions and properties of group rings which are pertinent for Schur rings. Section 2.2 will introduce the definition of Schur rings and will present many examples of Schur rings over finite groups. It will also present general constructions of Schur rings, including orbit and dot product Schur rings. This section also contains a collection of elementary properties of Schur rings which are fundamental for calculations in such rings. Most of these elementary properties were known and proven by Wielandt [35]. Section 2.3 focuses on Cayley maps, these being maps on group algebras which are induced from group homomorphisms. It also provides criteria for when the Cayley image of a Schur ring is also a Schur ring. Section 2.4 will generalize the notion of Schur rings in two ways: immersed Schur rings and pre-schur rings. Both types of rings naturally arise while studying Schur rings and deserve proper attention. Also many properties of Schur rings naturally extend to immersed Schur rings and pre-schur rings. Certain examples are also included here, including inflated Schur rings. From here we develop the construction of wedge products and their generalizations. Wedge products provide a method of extending Schur rings of normal subgroups by Schur rings of quotient groups. A result by Leung and Man (Theorem 2.66) states that every nontrivial Schur ring over a finite cyclic group is constructible using the methods mentioned in this chapter. Unless otherwise specified, G will denote a finite group and F a field with characteristic 4

13 zero. Let Z n = z n denote the cyclic group of order n. 2.1 Group Rings Let F [G] denote the group algebra of G with coefficients from F. For α F [G], we will often denote the coefficient of the group element g in α by α g, that is, α = g G α gg with α g F. Definition 2.1. For any α F [G] with α = α g g, we define g G α = g G α g g 1. Similarly, if C G, then C = {g 1 g C}. Proposition 2.2. Let α, β F [G] and let a, b F. Then (a) α = α, (b) (aα + bβ) = aα + bβ, (c) (αβ) = β α, Any function : A A on an F -algebra A satisfying Proposition 2.2 is called an involution and an algebra equipped with an involution is called a -algebra. Thus, every group ring is a -algebra. Definition 2.3. Let C G. We define C = g C g F [G]. An element α F [G] is a simple quantity if α = C for some C G. If C =, then C = 0. Proposition 2.4. Let D C G be subsets and let H, K G be subgroups. 5

14 (a) (C) = C, (b) C D = C D, (c) H K = H K HK. Let δ : F [G] F, called the augmentation map, be the linear map given as g G α gg g G α g. Proposition 2.5. Let G be a finite group and let F be a field. Then G Z(F [G]) and for any α = g G α g g F [G], we have that αg = Gα = δ(α)g. Proof. Let h G. We claim first that hg = Gh = G, which follows from the straightforward computation hg = h g = hg = g G g G g = G. h 1 g G The last equality holds because if g ranges over all the elements of the group then h 1 g also ranges over all the elements. A similar computation shows that Gh = G, which proves the claim. This shows also that G Z(F [G]). To finish the proof, let α F [G] and we compute ( ) αg = α g g G = g gg) g G(α g G = ( ) g G) = α g G = δ(α)g. g G(α g G Definition 2.6. Define a binary operation : F [G] F [G] F [G] as follows: if α, β F [G] with α = g G α g g and β = g G β g g, then α β = g G(α g β g )g. The operation is referred to as the Hadamard product or the circle product on F [G]. Proposition 2.7. Let α, β, γ F [G], let r F, h G, and C, D G. Then 6

15 (a) α (β γ) = (α β) γ, (b) α (β + γ) = α β + α γ and (α + β) γ = α γ + β γ, (c) G α = α G = α, (d) α β = β α, (e) (rα) β = α (rβ) = r(α β), (f) C D = C D. (g) C C = C, (h) C D = 0 if and only if C D =, (i) (α β) h = (α h) (β h), (j) (α β) = α β. Proof. Let α = g G α g g, β = g G β g g, and γ = g G γ g g. So, ( ) α (β γ) = α (β g γ g )g = g (β g γ g )]g g G[α g G = ( ) g β g ) γ g ]g = (α g β g )g γ = (α β) γ. g G[(α In summary, is associative because the multiplication of F is associative. This proves (a). A similar argument holds also for (b), (d), and (e). Next, G α = (1 α g )g = α g g = α. Similarly, α G = α, which gives (c). Suppose g G g G that α = C and β = D. Thus, α g = 1 if g C and α g = 0 if g / C. Similarly, β g = 1 if g D and β g = 0 if g / D. Thus, α g β g = 1 if and only if g C D and α g β g = 0 if and only if g / C D. Therefore, (f) holds. Properties (g) and (h) are immediate consequences of (f). Next, (α h) (β h) = which prove (i). Lastly, g G ( ) ( ) α g gh β g gh = g β g )gh = (α β) h, g G(α g G g G ( ) (α β) = (α g β g )g = g β g )g g G g G(α 1 = α g g 1 β g g 1 = α β, g G g G 7

16 which proves (j). Note that the previous proposition shows that (F [G], +, ) is always a commutative F - algebra with unity G. In fact, it is easy to check that (F [G], +, ) is isomorphic to the G -fold direct product of F. Furthermore, we have shown that (F [G], +, ) is a product of fields. Hence, (F [G], +, ) is semisimple. The proposition also shows that (F [G], +,, ) is a -algebra. We say that two simple quantities C and D are disjoint if C D = 0. In light of Proposition 2.7, C and D are disjoint if and only if C and D are disjoint. Definition 2.8. Let α F [G], such that α = g α gg. Let supp(α) = {g G α g 0}, which is called the support of α. For a simple quantity, supp(c) = C. We end the section by proving a result due to Wielandt, which shows that all -subalgebras of Q[G] are semisimple. In particular, Schur rings will be semisimple. Theorem 2.9 (Wielandt [34]). Every subalgebra of Q[G] which is closed under is semisimple. Proof. Suppose that S is a -subalgebra of Q[G] but not semisimple. Let J (S) denote the Jacobson radical of S. By Theorem A.12, J (S) 0 and contains a simple left ideal Sα, since S is artinian. But α J (S). So α(sα) = 0, and hence αα α = 0. Then αα αα = (αα )(αα ) = 0. We now claim that the only solution β Q[G] to the equation ββ = 0 is 0 itself. Suppose β = g β gg. Then the coefficient of 1 in ββ is g β2 g. Now, a sum of squares is 0 in Q if and only if β g = 0 for all g G. Thus, ββ = 0 implies that β = 0. By the above claim, it must be that αα = 0. Again using the claim, we conclude that α = 0, which contradicts J (S) 0. Therefore, S is semisimple. Theorem 2.9 is also true for all fields F R with the same proof. The result is also true for F = C with the same proof, although we must redefine the involution as α = g G α gg 1, where α g denotes the complex conjugate of α g. 8

17 2.2 Schur Rings Definition 2.10 (Schur Ring). Let {C 1, C 2,..., C r } be a partition of a finite group G and let S be the subspace of F [G] spanned by C 1, C 2,..., C r. We say that S is a Schur Ring over G if (i) C 1 = {1}; (ii) for each i, there is a j such that Ci = C j ; r (iii) for each i and j, we have C i C j = λ i,j,k C k, for constants λ ijk F. k=1 In the above equation, the λ i,j,k are referred to as the structure constants of S. For a Schur ring S over G, let D(S) = {C 1, C 2,..., C r } denote the partition corresponding to S. We will refer to the sets C 1,..., C r as the S-classes or the primitive sets of S. We also say that S is the Schur ring afforded by the partition D(S). Finally, the simple quantities C 1, C 2,..., C r in S will be referred to as the class sums of S. In summary of Definition 2.10, a subalgebra S F [G] is a Schur ring if it spanned by a basis of disjoint simple quantities, contains 1 and G, and is closed under. Notice that 2.10 (iii) implies that if a partition of G affords a Schur ring, then the product of any two primitive sets is a union of primitive sets. With respect to the Hadamard product, the class sums of a Schur ring S form an orthogonal basis of primitive central idempotents and G acts as unity. These properties characterize Schur rings. Theorem 2.11 ([24] Lemma 1.3). Suppose that S is a subalgebra of F [G]. Then S is a Schur ring if and only if S is closed under both and and contains both 1 and G. Proof. Suppose first that S is a Schur ring with partition D(S) = {C 1, C 2,..., C r }. By definition, S is closed under and 1, G S. So we need only show that S is closed under. To see this we compute, using Proposition 2.7, ( r ) ( r ) α β = α i C i β i C i = i=1 i=1 r α i β i (C i C i ) = i=1 r α i β i C i S, i=1 9

18 which proves the first direction. Next, suppose that S is closed under and. Now, consider the ring structure S = (S, +, ). Then S is a subalgebra of F [G] = (F [G], +, ). Clearly, F [G] is isomorphic to a G -fold product of F. Thus, F [G] is commutative and semisimple. Since F [G] is commutative, every subalgebra of F [G] is commutative and semisimple, including S. Therefore, there exists pairwise-orthogonal primitive idempotents τ i S such that S = (S τ 1 ) (S τ 2 )... (S τ r ). (2.1) Since τ i τ i = τ i, it must be that τ i is a simple quantity, that is, there exist some C i G such that τ i = C i. Since τ i τ j = 0 for i j, we have C i C j =. The primitivity of τ i requires that S τ i is a field extension of F contained in g C i (F [G] g). Since F [G] g = F for each g G, it must be that S τ i = F for each i. Equation (2.1) then shows that {C 1, C 2,..., C r } is a F -basis for S. Next, since S is closed under, the involution is a ring automorphism S S satisfying (α β) = α β by Proposition 2.7. Thus, (C i ) is also a primitive idempotent. Since {C 1, C 2,..., C r } contains all the primitive idempotents of S, (C i ) = C j for some j. If additionally G S, then {C 1, C 2,..., C r } must also form a partition of G. Lastly, if 1 S, then S is a Schur ring over G. Lemma Let S be a Schur ring over G. Let g G such that {g} D(S). Then gc, Cg D(S) for all C D(S). Proof. Let D(S) = {C 1, C 2,..., C r }. Then gc i = k λ kc k for S-classes C k. Now, g 1 = g S. Thus, C i = k λ kg 1 C k. Since C k C j = implies that g 1 C k g 1 C j =, we see that C i = g 1 C k for some k. Therefore, gc i = C k. Proposition Let S be a Schur ring over G. Let H = {h {h} D(S)}. Then H G. Proof. Clearly, 1 H. Also, if {h} D(S), then {h 1 } = {h} D(S). So, H is closed under inverses. Lastly, if {g}, {h} D(S), then {gh} D(S) by the previous lemma. So, H is closed under multiplication and hence is a subgroup of G. 10

19 We next provide a few examples of Schur rings. Example Every finite group algebra F [G] is a Schur ring, where each class of D(F [G]) consists of only a single element. For this reason, Schur rings may be thought of as a generalization of group rings. Naturally, F [G] is the largest possible Schur ring over G, that is, it is the unique Schur ring which contains all other Schur rings of G. Example At the other extreme, consider the partition G = {1} (G {1}) and let S be the subring of F [G] generated by these two class sums. Certainly, 1 = 1 and (G 1) = G 1. Also, (G 1) 2 = G 2 2G+1 = ( G 2)G+1 = ( G 2)(G 1)+( G 1). Therefore, S is a Schur ring, which we refer to as the trivial Schur ring. The trivial Schur ring is always contained in the center of F [G] and hence is a commutative ring, even if G is nonabelian. The trivial Schur ring will be denoted as F [G] 0. A complete multiplication table of F [G] 0 can be found in Table 2.1. Table 2.1: Multiplication Table for F [G] 0 τ 1 = 1 τ 2 = G 1 τ 1 τ 1 τ 2 τ 2 τ 2 ( G 1)τ 1 + ( G 2)τ 2 The trivial Schur ring F [G] 0 is the unique Schur ring of F [G] of smallest dimension, that is, F [G] 0 is the unique Schur ring contained in all Schur rings over G. When G 1, F [G] 0 is the unique Schur ring of dimension 2. Example Let G = S 3, the symmetric group on 3 elements. Let D = {{1}, {(12)}, {(123), (321)}, {(13), (23)}}. This partition of G affords a Schur ring as shown in Table 2.2. Example Let G = Z 7 = z, the cyclic group of order 7. Let D = {{1}, {z, z 2, z 4 }, {z 3, z 5, z 6 }}. 11

20 Table 2.2: Multiplication Table for the Schur Ring in Example 2.16 τ 1 = 1 τ 2 = (12) τ 3 = (123) + (321) τ 4 = (13) + (23) τ 1 τ 1 τ 2 τ 3 τ 4 τ 2 τ 2 τ 1 τ 4 τ 3 τ 3 τ 3 τ 4 2τ 1 + τ 3 2τ 2 + τ 4 τ 4 τ 4 τ 3 2τ 2 + τ 4 2τ 1 + τ 3 This partition of G generates a Schur ring as shown in Table 2.3. Table 2.3: Multiplication Table for the Schur Ring in Example 2.17 τ 1 = 1 τ 2 = z + z 2 + z 4 τ 3 = z 3 + z 5 + z 6 τ 1 τ 1 τ 2 τ 3 τ 2 τ 2 τ 2 + 2τ 3 3τ 1 + τ 2 + τ 3 τ 3 τ 3 3τ 1 + τ 2 + τ 3 2τ 2 + τ 3 Example Let H G and let S be the subspace of F [G] afforded by the partition D(S) = {{1}, H {1}, G H}. In particular, S = Span F 1, H 1, G H = Span F 1, H, G. Notice that H H = H H, H G = G H = H G, and G 2 = G G. Also, H = H. Theorem 2.11 then shows that S is a Schur ring. This kind of Schur ring is our first example of a wedge product of Schur rings and is the simplest kind of wedge product. Wedge products are defined later in Example Example 2.19 (Lattice Schur Rings). Let G be a finite group and L be a sublattice of the lattice of normal subgroups of G. Then we define S(L) = Span F {H H L}. 12

21 Since H K = H K and H K = H K HK for H, K G, S(L) is a Schur ring, by Theorem For this reason, S(L) will be called a lattice Schur ring. It should be mentioned that D(S) {H : H L}. For any finite group G, the trivial Schur ring is a lattice Schur ring, corresponding to the lattice {1, G}. The Schur ring from Example 2.18 (in the case that H G) is another example of a lattice Schur ring, using the lattice {1, H, G}. Example 2.20 (Orbit Schur Rings). Let H Aut(G). Let F [G] H = {α F [G] σ(α) = α, for all σ H}, that is, it is the largest subring of F [G] which is fixed by the automorphism group H. We claim that F [G] H is a Schur ring of G. By Theorem B.3, F [G] H is an F -subalgebra of F [G] with unity that is generated by the periods of the elements of G with respect to H. In particular, F [G] H has a basis of disjoint simple quantities whose sum is G. Since σ(g 1 ) = σ(g) 1, we have that C D(F [G] H ) for all primitive sets C. Therefore, F [G] H is a Schur ring, as claimed, whose partition of G is the H-orbits of G. The Schur ring F [G] H is called an orbit Schur ring. The group ring F [G] is an orbit Schur ring with respect to H when H = 1 Aut(G), that is, F [G] = F [G] Id. Example 2.21 (Rational Schur Rings). Let R(F [G]) = F [G] Aut(G), that is, the Schur ring whose partition is the automorphism classes of G. Any Schur ring contained in R(F [G]) is called a rational Schur ring since it is fixed by all group automorphisms. Rational Schur rings have been well studied in the literature, especially in the case of cyclic groups. It was observed by Muzychuk in [23] that for cyclic groups the lattice Schur rings correspond exactly with the rational Schur rings. Understanding the rational Schur rings is useful in developing structure theorems of Schur rings over cyclic groups [15, 23]. Example 2.22 (Central Schur Rings). Let G be any finite group and consider S = F [G] Inn(G), where Inn(G) is the group of inner automorphisms of G. So, S is a Schur ring whose partition D(S) is the collection of conjugacy classes. In fact, S = Z(F [G]), since an element α Z(F [G]) if and only if gα = αg for all g G if and only if g 1 αg = α 13

22 for all g G if and only if α F [G] Inn(G). Therefore, the center of a group ring is always a Schur ring. Any Schur ring contained in Z(F [G]) is called a central Schur ring. The importance of the structure of the group ring F [G] and its center Z(F [G]) is readily seen in representation theory. Example 2.23 (Symmetric Schur Rings). Let G be a finite group. Now, Aut(G) if and only if G is abelian. Thus, if G is abelian, the subalgebra F [G] is a Schur ring of G whose classes are of the form C g = {g, g 1 }. We will denote this Schur ring by S(F [G]). We say an element α of F [G] is symmetric if α = α. Thus, S(F [G]) is the collection of all symmetric elements of F [G]. Any Schur ring contained in S(F [G]) is called a symmetric Schur ring. For example, lattice Schur rings are always symmetric. Example When G is a nonabelian group, there is no guarantee that the collection of symmetric elements forms a subring of F [G]. For example, let G = S 3 and consider the symmetric elements of Q[G]. Since transpositions have order 2, each transposition in S 3 is its own inverse. Thus, S(Q[G]) contains all transpositions of G. But the product (12) (23) is a 3-cycle and not contained in S(Q[G]), since 3-cycles have order 3. So, S(Q[G]) is not a ring and hence not a Schur ring. On the other hand, let G = Q 8, the quaternion group of 8 elements. For Q 8, the inverse classes of G are the same as the conjugacy classes of G. Thus, S(Q[G]) is the center of Q[G], which is always a Schur ring. In particular, if C x denotes the conjugacy class of x G, then D(S) = {C 1, C -1, C i, C j, C k } and the multiplication table for the Schur ring is given in Table 2.4. Proposition Let S and T be Schur rings over G. Then S T is a Schur ring over G. Proof. Certainly, 1, G S T. If α S, then α S. Likewise, if α T, then α T. So, α S T whenever α S T. Lastly, suppose α, β S T, then α β S, T by Theorem This implies that α β S T. Therefore, S T is a Schur ring, again by Theorem The set of all partitions of a finite group G forms a lattice, defined with and given as follows: if P and Q are partitions of G, then P Q is the largest partition of G contained 14

23 Table 2.4: Multiplication Table of Z(Q[Q 8 ]) τ 1 = C 1 τ -1 = C -1 τ i = C i τ j = C j τ k = C k τ 1 τ 1 τ -1 τ i τ j τ k τ -1 τ -1 τ 1 τ i τ j τ k τ i τ i τ i 2τ 1 + 2τ -1 2τ k 2τ j τ j τ j τ j 2τ k 2τ 1 + 2τ -1 2τ i τ k τ k τ k 2τ j 2τ i 2τ 1 + 2τ -1 both in P and Q and P Q is the smallest partition of G which contains both P and Q. Proposition 2.25 then says that D(S T ) = D(S) D(T ). On the other hand, D(S) D(T ) does not afford a Schur ring in general. Example 2.26 (Dot Products). Let S and T be Schur rings over G and H, respectively. We naturally can view G and H as subgroups of G H. Let D = {CD C D(S), D D(T )}, (2.2) that is, D is the partition of G H generated by all the possible products of S- and T - classes. Let S T = Span F {CD C D(S), D D(T )} = Span F {C D C D(S), D D(T )}, the subspace of F [G H] afforded by D. Since both D(S) and D(T ) contain the identity class {1}, S T contains an isomorphic copy of S and T and they centralize each other in S T. Furthermore, 1 S T and G H = G H S T. For any S- and T -classes C and D, respectively, (C D) = D C = C D S T. Thus, S T is closed under. Lastly, if C 1, C 2 D(S) and D 1, D 2 D(T ), then C 1 C 2 S, D 1 D 2 T and (C 1 D 1 )(C 2 D 2 ) = (C 1 C 2 )(D 1 D 2 ) S T. 15

24 Therefore, S T is a Schur ring over G H. We refer to S T as the dot product Schur ring of S and T or the direct product Schur ring. By some authors, S T is denoted as S T. It is a fact that F [G H] = F [G] F F [H], as F -algebras. Using similar reasoning, it is true that S T = S F T, as F -algebras. Because of this isomorphism, S T is sometimes referred to as the tensor product Schur ring of S and T and denoted as S T. The Schur ring S T also has the property that it is the smallest Schur ring of G H which contains the subalgebras S and T and hence is the composite or join of the two Schur rings. Lemma Let G 1, G 2 be finite groups and H i Aut(G i ). Then Q[G 1 G 2 ] H 1 H 2 = Q[G 1 ] H1 Q[G 2 ] H 2. Proof. Let C be the automorphism class of (g 1, g 2 ) with respect to H 1 H 2. Let C i be the automorphism class of g i with respect to H i, i = 1, 2. Then (g 1, g 2 ) is automorphic to (g 1, g 2) under H 1 H 2 if and only if g 1 is automorphic to g 1 under H 1 and g 2 is automorphic to g 2 under H 2 if and only if C = C 1 C 2. The result then follows. We present now two more constructions of Schur rings which generalize the method of dot products from Example Example 2.28 (Central Products). Let G be a finite group. Let H, K G such that G = HK and H and K centralize each other, that is, [H, K] = 1. Then G = H Z K is the central product of H and K. As a consequence, H, K G. Let L = H K. Certainly, L Z(G). Let S and T be Schur rings over H and K, respectively, such that F [L] S T, that is, the restriction of S and T to the subgroup L is the whole group ring on L. Let S Z T = Span{CD C D(S), D D(T )}. We claim that S Z T is a Schur ring over G. 16

25 Let D = {CD C D(S), D D(T )}. (2.3) So, S Z T = Span F {B B D}. We first show that D forms a partition of G. Let g G. Since G = HK, there exists h H, k K such that g = hk. Now, there exists some C D(S) and D D(T ) such that h C and k D. Thus, g CD. Suppose next that there exists sets C 1, C 2 D(S) and D 1, D 2 D(T ) such that g C 1 D 1 C 2 D 2. In particular, there exists elements h i C i, k i D i such that g = h 1 k 1 = h 2 k 2. Then h 1 2 h 1 = k 2 k 1 1 H K = L. Since h 1 2 h 1 L, C 2 (h 1 2 h 1 ) D(S). This implies that C 2 (h 1 2 h 1 ) = C 1. Likewise, (k 2 k 1 1 ) 1 D 2 = D 1. Therefore, C 2 D 2 = C 2 (h 1 2 h 1 )(k 2 k 1 1 ) 1 D 2 = C 1 D 1. Therefore, D forms a partition on G. Since {1} D(S) D(T ), {1} D. Next, if CD D, then (CD) = D C = C D D since C D(S) and D D(T ). To show that S Z T is a Schur ring, it remains to prove that S Z T is a ring. For this purpose, we will first show that C D = µcd, for some positive integer µ. To prove this claim, consider H K = C D = C D (2.4) C,D C D(S) D D(T ) = H K HK = L B D B. (2.5) Clearly, supp ( C D ) CD, and, by construction, every B D is of the form B = CD for some C D(S) and D D(T ). By comparing coefficients in (2.4) and (2.5), we get L B = C D. CD=B Suppose that C 1 D 1 = B = C 2 D 2. Then there exists some l L such that C 1 = C 2 l and D 1 = l 1 D 2, by the work done above. Thus, C 2 D 2 = (C 2 l) (l 1 D 2 ) = C 2 l l 1 D 2 = C 1 D 1. 17

26 Therefore, if n is the number of terms in the sum CD=B C D, then L CD = n(c D). Then the previous equation implies that C D = L CD = µcd. n Since the coefficients of each group element in C D necessarily are positive integers, this proves the claim. Since S and T are Schur rings, there exists structure constants λ ijk and κ rst such that C i C j = k λ ijk C k and D r D s = t κ rst D t. Then ( ) ( ) ( ) ( ) ( ) ( ) Ci D r Cj D s = Ci C j Dr D s = λ ijk C k κ rst D t k t = (λ ijk κ rst ) C k D t = (λ ijk κ rst µ kt ) C k D t S Z T. k,t k,t Therefore, S Z T is a Schur ring, which we refer to as the central product Schur ring of S and T. By comparing equations (2.2) and (2.3) and recognizing that H Z K = H K when L = 1, we note that S Z T generalizes the construction in Example Thus, we may also denote S Z T as S T. Example 2.29 (Semi-direct Products). Let G be a finite group. Let H G, K G such that G = HK and H K = 1. Then G = H K is the semi-direct product of H and K. As a consequence, conjugation of K on H induces a homomorphism ϕ : K Aut(H). Let S and T be Schur rings over H and K, respectively, such that S F [H] ϕ(k) = S, that is, S is ϕ(k)-rational. Let S T = Span{CD C D(S), D D(T )}. 18

27 We claim that S T is a Schur ring over G. Let D = {CD C D(S), D D(T )}. Since H K = 1, we see D is a partition of G. Since S is ϕ(k)-rational, the classes of S commute with the classes of T in F [G]. So, as in the case of direct and central products, 1, G, (CD) S T for all C D(S) and D D(T ). Thus, it remains to prove that S T is a ring. But as in the case of direct products, CD = C D = D C. Thus, S T is closed under multiplication. Therefore, S T is a Schur ring, which we refer to as the semi-direct product Schur ring of S and T. Like the last example, S T also generalizes the construction in Example Thus, we may also denote S T as S T. We now will end this section by proving some elementary propositions about Schur rings that will be useful in future proofs. All of these results are due to Wielandt [35] and will be built upon a fundamental lemma of Schur rings, Lemma This lemma is preceded by a definition. Definition Let S be a Schur ring over G and let C G. We say that C is an S-set of G if C S. If C is an S-set and a subgroup of G, then we say that C is an S-subgroup of G. The following is clear. Lemma Let G be a finite group and let S be a Schur ring over G. Let α S such that α = g G α gg. Then {g G α g = c} is an S-set for each c F. We now begin with the first of the propositions. Proposition Let S be a Schur ring over G and let α = g G α gg S. Then supp(α) is an S-set. Proof. Let K c = {g G α g = c} for c F. By Lemma 2.31, K c S for all c F. Then supp(α) = K αg = K αg S. g supp(α) α g:g supp(α) Therefore, supp(α) is an S-set. 19

28 Proposition Let S be a Schur ring over G, let α S, and let H = supp(α). Then H S. In particular, if C D(S), then C is an S-subgroup. Proof. Let L = supp(α). Since H = L is finite, there exists some integer n sufficiently large such that H = n i=1 Li. Since char F = 0, H = supp Proposition ( n i=1 Li), which is an S-set by Proposition Let S be a Schur ring over G, let α = g G α gg S, and let f : F F be any function. Then f[α] = g G f(α g)g S. Proof. Let K c = {g G α g = c}. Then K c S for each c F. Now, if α = ck c, then f[α] = f(c)k c S. Proposition Let S be a Schur ring over G. Let α S and Stab(α) = {g G αg = α}. Then Stab(α) is an S-subgroup of G. Proof. Let α = g G α gg S, let K c = {g G α g = c} for c F, and let M c = {g G K c g = K c }, that is, M c is the subset of G which permutes K c. Let g M c. Then there exists K c many solutions (h, k) K c K c to the equation hg = k. But each solution is also a solution to the equation g = kh 1. Thus, the coefficient of g in K c K c is Kc. Conversely, if the coefficient of g in K c K c is Kc, then there are K c distinct solutions (h, k) K c K c to g = kh 1, i.e. hg = k. Thus, K c g = K c and g M c. Then applying Lemma 2.31 to K c K c and the coefficient Kc, we conclude that M c S. Now, Stab(α) = g supp(α) M αg = M αg S. g supp(α) Proposition Let G be an abelian group, let S be a Schur ring over G, and let α = g α gg S. Define α (m) = g α gg m for m Z. Then α (m) S for every integer m coprime to G. Furthermore, define C (m) = {g m g C} for each C G and m Z. Then if C D(S), then C (m) D(S) for every integer m coprime to G. Proof. Clearly, the map (m) : F [G] F [G] is linear, so it suffices to prove the statement for a simple quantity α. Also, we note that α ( 1) = α and α (mm ) = (α (m) ) (m ). Thus, it suffices to prove the statement for m = p, a prime number not dividing G. 20

29 Since α is simple, there exists some subset C G such that α = C. Since G is abelian, F [G] is a commutative ring and the polynomial congruence ( ) p g g p (mod p) g C g C holds. Let f p : F F be the function defined as n mod p, n Z f p (n) = 0, otherwise. Thus, f p [α (p) ] = f p [α p ] S by Proposition Now, when p G, the map g g p is a group automorphism. So, α (p) is a simple quantity and α (p) = f p [α (p) ] S. Let C be a primitive set of S. By the above, we see that C (m) is an S-set. If C (m) is not primitive, then let D be one of the primitive subsets of C (m). In particular, D < C (m) = C. Let 1 = am + b G for some integers a, b. Then D (a) is an S-set, but D (a) C, which contradicts C being primitive. Therefore, C (m) must also be primitive. Proposition Let G be a cyclic group and let S be a Schur ring over G. Let σ Aut(G) and α S. Then σ(α) S. In particular, if C D(S), then σ(c) D(S). Proof. Since every automorphism σ is of the form g g m for some integer m relatively prime to G, the result follows immediately from the previous proposition. Definition A Schur ring S over a finite group G is primitive if the only S-subgroups are 1 and G. For primitive Schur rings, every non-trivial primitive set necessarily generates the whole group. The trivial Schur ring is a typical example of a primitive Schur ring. As Wielandt has shown, for many abelian groups, this is the only example. Theorem 2.39 (Wielandt). If G is a finite abelian group not of prime order with a nontrivial, cyclic Sylow subgroup, then the only primitive Schur ring over G is the trivial Schur ring. 21

30 Proof. Its proof can be found in [35] or in [29, Theorem ]. Example Let G = Z 3 Z 3 = a, b and let S = Span Q {1, a + a 2 + b + b 2, ab + a 2 b 2 + ab 2 + a 2 b}. Then S is an orbit Schur ring afforded by the automorphism subgroup generated by the automorphism σ : a b, b a 2. Now, the set of S-subgroups is simply {1, G}, that is, S is primitive. Of course, G has no nontrivial, cyclic Sylow subgroup. The multiplication table of S is shown in Table 2.5. Table 2.5: Multiplication Table for the Schur Ring in Example 2.40 τ 1 = 1 τ 2 = a + a 2 + b + b 2 τ 3 = ab + a 2 b 2 + ab 2 + a 2 b τ 1 τ 1 τ 2 τ 3 τ 2 τ 2 4τ 1 + τ 2 + 2τ 3 2τ 2 + 2τ 3 τ 3 τ 3 2τ 2 + 2τ 3 4τ 1 + 2τ 2 + τ 3 When G = Z p, for some prime p, every Schur ring is necessarily primitive. These Schur rings will be considered in Theorem Cayley Maps Now that we have developed many of the elementary properties of Schur rings, it is natural next to compare Schur rings via homomorphisms. Seeing that Schur rings are subalgebras of F [G], it is natural to relate them via ring or algebra homomorphisms. An algebra homomorphism is a map which preserves the ring and linear structure of the Schur ring. More specifically, let S and T be Schur rings over G and H, respectively, and let ϕ : S T be an algebra homomorphism. Let α, β S and let c F. Then ϕ(α + β) = ϕ(α) + ϕ(β), 22

31 ϕ(cα) = cϕ(α), ϕ(α β) = ϕ(α) ϕ(β). Thus, ϕ(s) is a subalgebra of T. Unfortunately, algebra homomorphisms are not sufficient to study Schur rings. For example, for any two finite groups G and H, we have F [G] 0 = F [H] 0 ( ) as F -algebras. To see this, note that F [G] 0 = (G) 1 1 G as simple ideals. Now, G (G) ( ) = 1 1 G = G F, as F -algebras, independent of the group G. For another example, consider the group algebras C[Z 4 ] and C[Z 2 Z 2 ]. As C-algebras, C[Z 4 ] = C[Z 2 Z 2 ] = C 4. Thus, the appropriate homomorphisms of Schur rings need to be stronger than mere algebra homomorphisms. Although ϕ(s) is an algebra, by Theorem 2.11, ϕ(s) needs to also be closed under and in order to be a Schur ring. An arbitrary algebra homomorphism need not preserve these two additional operations, as illustrated above. Thus it is natural to define a Schur homomorphism to be a linear map ϕ : S T between Schur rings S and T such that: ϕ(α β) = ϕ(α) ϕ(β), ϕ(α β) = ϕ(α) ϕ(β), ϕ(α ) = ϕ(α), for all α, β S. If ϕ : S T is also bijective, then ϕ is a Schur Isomorphism. An immediate consequence of this type of homomorphism is that the image ϕ(s) is a Schur ring over some T -subgroup, specifically supp(ϕ(g)). Schur homomorphisms were studied by Muzychuk in [24], in which it was proven that Schur rings over a cyclic group are Schur isomorphic if and only if they coincide. Tamaschke also considered Schur homomorphisms 2.1 in his attempt to define a category of Schur rings in [31]. These are only a few examples from the literature. Suppose S is a Schur ring over G. Considering the ring structure S = (S, +, ), S = Q n, where n = dim S, and hence S is semisimple. Thus, any homomorphism ϕ : S T is simply a function from the S-classes into the T -classes. Thus, if ϕ is a Schur isomorphism, 2.1 Tamaschke s original definition of a homomorphism of Schur algebras differs from our presentation, although the two definitions are equivalent. 23

32 then ϕ induces a bijection between the primitive sets of S with the primitive sets of T. From a categorical sense, this class of morphisms is appropriate; that is, Schur maps are exactly the homomorphisms which preserve the operations of Schur rings. On the other hand, Schur rings were originally used to study groups and much of the algebraic structure of Schur rings depends on the group, so it would be useful for the homomorphisms of Schur rings to also relate to the group. It is possible for nonisomorphic groups to have Schur isomorphic Schur rings, e.g. Z 8 and D 4, the dihedral group of order 8, both have isomorphic Schur rings of dimension three. Instead of Schur homomorphisms, we will study maps which preserve the group structure of Schur rings. These maps will be algebra homomorphisms but will not necessarily preserve Hadamard multiplication. Under certain circumstances, the image of a Schur ring will be a Schur ring. Thus, these maps will sometimes provide a class of more useful homomorphism of Schur rings. We introduce now the notion of a Cayley map. Definition Let G and H be groups, let A and B be subalgebras of F [G] and F [H], respectively, and let f : A B be an F -algebra homomorphism. If there exists an F - algebra homomorphism ϕ : F [G] F [H] such that ϕ A = f and ϕ G : G H is a group homomorphism, then we say that f is a Cayley homomorphism. A bijective Cayley homomorphism is a Cayley isomorphism. For example, let G be a group, let H be a subgroup of G, and let σ Aut(G). Then the group algebras F [H] and F [σ(h)] are Cayley isomorphic in F [G]. If S is a Schur ring over G, then S and σ(s) are Cayley isomorphic. Let ϕ : G H be a group homomorphism. Let ϕ also denote its linear extension ϕ : F [G] F [H]. Let g G. Then ϕ(g ) = ϕ(g 1 ) = ϕ(g) 1 = ϕ(g). By linearity, ϕ(α ) = ϕ(α) for all α F [G]. In particular, a Cayley map is a -algebra homomorphism, that is, Cayley maps always preserve the involution structure of F [G]. Likewise, Cayley maps preserve the involution structure of any -subalgebra of F [G], including Schur rings. Let δ : F [G] F be the augmentation map, which is a Cayley map induced from the 24

33 trivial map δ : G 1, and let α F [G], with α = g α gg. Then ( ) δ(α α) = δ αgg 2 = αg 2 g G g G and ( ) ( ) ( ) 2 δ(α) δ(α) = α g α g = α g. g G g G g G This shows that δ does not preserve Hadamard products in general and that Cayley maps are typically not Schur homomorphisms. The next proposition determines a necessary and sufficient condition for when a Cayley map is a Schur map. Proposition Let ϕ : F [G] F [H] be a Cayley map. Then ϕ is injective if and only if ϕ(α β) = ϕ(α) ϕ(β) for all α, β F [G]. Proof. Suppose that ϕ is injective. Let α = g G α gg and β = g G β gg. Then ( ) ϕ(α β) = ϕ α g β g g = α g β g ϕ(g) g G = g G ( ) ( ) α g ϕ(g) β g ϕ(g), since ϕ(g) occurs only once in the above sum, g G g G ( ) ( ) = ϕ α g g ϕ β g g = ϕ(α) ϕ(β). g G g G Conversely, suppose that ϕ is not injective. Let K = ker(ϕ G ) 1. Then ϕ(g G) = ϕ(g) = K ϕ(g) K 2 ϕ(g) = K 2 (ϕ(g) ϕ(g)) = ( K ϕ(g)) ( K ϕ(g)) = ϕ(g) ϕ(g), which proves the remaining direction. Corollary Every Cayley isomorphism is a Schur isomorphism. Cayley isomorphic is a strictly stronger condition than Schur isomorphic. For example, 25

34 consider the partition {1}, {z 4 }, {z, z 2, z 3, z 5, z 6, z 7 } over Z 8 = z. This partition affords a Schur ring over Z 8, which we denote as S. This Schur ring is Schur isomorphic to the Schur ring T over D 4 associated to the partition {1}, {s}, {r, r 2, r 3, rs, r 2 s, r 3 s}. Here D 4 is the dihedral group of order 8, D 4 = r, s r 4, s 2, s 1 rs = r 1. Now, if S is Cayley isomorphic to T, then there exists some group homomorphism ϕ : Z 8 D 4 such that ϕ(s) = T. Since ϕ(s) contains D 4, ϕ must be surjective. Considering the orders of the groups, ϕ must also be injective, that is, ϕ : Z 8 which is absurd. D 4 is a group isomorphism, Therefore, S and T are Schur isomorphic but not Cayley isomorphic. In particular, Schur isomorphic Schur rings associated to nonisomorphic groups cannot be Cayley isomorphic by this same argument. The following formula was proven in [24]. Proposition Let ϕ : G H be a group homomorphism with ker ϕ = K. Let α, β F [G]. Then ϕ(α) ϕ(β) = 1 ϕ((α K) (β K)). K Proof. Suppose that α = g α gg and β = g β gg. Then, the left hand side is The right hand side is ( ) ( ) ϕ(α) ϕ(β) = ϕ α g g ϕ β g g = g G h ϕ(g) = h ϕ(g) ϕ(g)=h ϕ(g)=h α g α g g G h ϕ(g)=h h ϕ(g) β g h. ϕ(g)=h β g h [( ( ) ) ( ( ) )] 1 1 ϕ((α K) (β K)) = K K ϕ α g g β g g g G g g K g G g g K 26

35 [ = 1 K ϕ = K K g G h ϕ(g) ( g g K ϕ(g)=h α g ) ( α g g g K β g ) ϕ(g)=h β g g ] h. Corollary Let ϕ : G H be a group homomorphism with ker ϕ = K. Let S be a Schur ring over G such that K S. Then ϕ(s) is a Schur ring over a subgroup of H. Furthermore, if ϕ is surjective, then ϕ(s) is a Schur ring over H. Proof. It is always the case that ϕ(s) is a -subalgebra of F [H] for any Schur ring over G without further assumption. By Proposition 2.44, ϕ(s) is closed under. Thus, ϕ(s) is a Schur ring over ϕ(g) by Theorem Corollary 2.45 was originally proved by Leung and Ma [15] using a different proof. As an example, if G = H 1 H 2, π 1 : G H 1 and π 2 : G H 2 are the canonical projections, and S 1 and S 2 are Schur rings over H 1 and H 2, respectively, then π i (S 1 S 2 ) = S i for i = 1, 2. As stated earlier in Example 2.19, for any lattice L of normal subgroups of a finite group G, S(L) is a Schur ring over G spanned by the elements of L. Let N G and let ϕ : G G/N be the quotient map. Suppose that L is a distributive lattice, that is, A (BC) = (A B)(A C) and A(B C) = (AB) (AC), for all A, B, C L. Then we claim that ϕ(s(l)) is a Schur ring over G/N, even if N / L. As in the proof of Corollary 2.45, it suffices to show that ϕ(s(l)) is closed under. K 1, K 2 L, then K 1 N K 2 N = K 1 N K 2 N = (K 1 K 2 )N = = 1 K 1 K 2 N (K 1 K 2 ) N, 1 (K 1 K 2 ) N K 1 K 2 N If 27

36 where the second equality holds by the distributivity of the lattice. Then ϕ(k 1 ) ϕ(k 2 ) = 1 N ϕ((k 1 N) (K 2 N)), by Proposition 2.44, = K 1 N K 2 N ϕ(k 1 N K 2 N) N = K 1 N K 2 N N K 1 K 2 N ϕ((k 1 K 2 ) N) = (K 1 N)(K 2 N) ϕ(k 1 K 2 ) ϕ(s(l)). Therefore, ϕ(s(l)) is closed under, which proves the claim. This fact is reported in the next proposition. Proposition Let G be a finite group and let L be a distributive lattice of normal subgroups of G. Let ϕ : G H be a group homomorphism. Then ϕ(s(l)) is a lattice Schur ring over a subgroup of H. Let G be a finite cyclic group. Then the lattice of subgroups of G is distributive, and hence any sublattice is also distributive. Thus, ϕ(s(l)) is a Schur ring for any group homomorphism ϕ and any lattice L of subgroups of G. Theorem 2.68 will generalize this for any Schur ring over a cyclic group. On the other hand, the Cayley image of a Schur ring need not be a Schur ring. In fact, it is false even for Schur rings over abelian groups, as illustrated in the following example. Example Let G = Z 2 Z 6 = a, b and let S = Span Q {1, b 3, b 2 + b 4, b + b 5, a + ab 3, ab + ab 2, ab 4 + ab 5 }. Then S is an orbit Schur ring afforded by the subgroup generated by the automorphism σ : a ab 3, b b 1. Let ϕ : G Z 6 be the projection homomorphism onto the subgroup b, that is, π : a 1, b b. Then ϕ(s) = Span Q {1, b 3, b 2 + b 4, b + b 5, 1 + b 3, b + b 2, b 4 + b 5 } = Span Q {1, b 3, b 2 + b 4, b + b 5, b + b 2 }. 28